Research Areas

My research is  focused on the analysis of the nonlinear PDEs with the primary focus on  equations arise in fluid dynamics and wave theory. I am mainly interested in the existence, uniqueness and regularity of the PDEs arise in this field.   I also address some quantitative properties of solutions to these kind of equations, namely: long time behavior as well as stability.   In my research, I apply a wide range of advanced techniques in harmonic analysis and PDE's, including Littlewood--Paley theory.  In addition, I make use of the methods in dynamical systems, and my recent works on both decay rates and stability of PDE's rely heavily on the spectral theory.

 In this context, I have worked on the Navier-Stokes systems and wave theory. Since their derivation, many physical phenomena are modeled by wave equations and  Navier-Stokes systems. Navier-stokes systems are usually derived by coupling the Navier-Stokes equations with other equations. For example the Magnetohydrodynamics (MHD) system consists of a coupling of the Navier-Stokes equation with the Maxwell's equation, and models the magnetic properties of electrically conducting fluids, including plasma, liquid metals, electrolytes, etc. 

Projects in preparation

1. Global regularity for the 2D Magneto-Micropolar Equation in the super critical region

2. With With J. Kelliher and D. Ambrose:  Surface Quasi_geostrophic Patches and Layers. 

3. With D. Ambrose: Global Well-posedness and long time behavior of the coordinate free model of the flame front. 

4.  With D. Wright: The dynamic of diatomic FPU system toward monotonic FPU system.  

Publications

1. With A. Stefanov: Sharp Asymptotics for global solutions of the parabolic Hardy-Sobolev equation.  (Submitted) 

2.  With H. Mofidi:  On the sharp time decay rates for the fractional Boussinesq system. (Submitted) 

3.  With J. Kelliher and D. Ambrose:  Generalized Surface Quasi_geostrophic Patches and Layers.  (To appear in Nonlinearity)

4. With H. Mofidi:  Analysis of Flux Ratios Bifurcation in Ionic Flows via Classical Poisson-Nernst-Planck Models, Studies in Applied Mathematics (2025)

5. With David Ambrose and J. Kelliher: Periodized solutions to the 2D Euler equations, periodic arrays of vortex patches, and periodic vortex layers. Siam J. on Mathematical Analysis (2024).

6.  With F. Farivar: A Chemotaxis model of Prey and Predator model of three species: Existence of global solutions., JMAA (2024)

7.   F. Hadadifard,  Optimal decay rates and attractors of  the 2D fully dissipative Magneto Hydrodynamics system., ZAMP, 72, Article number: 132 (2021).

8.    With D. Ambrose,  D. Wright, Well-posedness and asymptotics  of a coordinate-free model of flame fronts.  Siam J. on Applied Dynamical system, (2021).

9.   With A. Stefanov, Steady states of the weakly forced SQG equation: existence and sharp relaxation rates.   J. Math. Fluid Mechanics, (2021).

10.  With D. Wright,  Mass-in-Mass Lattices with Small Internal Resonators. Studies in Applied Mathematics, (2021).

11.   With A. Stefanov, Sharp relaxation rates for plane wave of general reaction-diffusion models. J. Math. Phys. Vol.61, Issue 4, (2020).

12. With S. Malhi, Z. Xiao,  C_0-Semigroup Numerical Approximation to Damped Wave equations., Journal of Numerical Analysis, Industrial and Applied Mathematics, ( 2020).

13.   (Ph.D Thesis) With A. Stefanov, Sharp time asymptotics for the quasi-geostrophic equation, the Boussinesq system and near plane waves of reaction-diffusion models. 

14. With A. Stefanov, On the sharp time decay rates for the 2D generalized Quasi-Geostrophic equation and the Boussinesq system.  Journal of Nonlinear Science, 29(5), (2019), pp 2231-2296.   

15.  With A. Stefanov, On the global regularity of the 2D critical Boussinesq system with $\alpha > 2/3$, Comm. Math. Sci., 15(2017), p. 1325-1351. 

16.   With H.Mofidi, Weight guaranteeing polefree barycentric rational interpolation, Indian J. of Science and Technology, 6(11) (2013), p. 5450-5458.